Research Article

Received 30 March 2009 Published online 15 July 2009 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/mma.1198MOS subject classification: 35 Q 35; 76 D 03

Global well-posedness for twomodified-Leray-a-MHD models with partialviscous terms

Yong Zhoua∗† and Jishan Fanb

Communicated by W. Sprößig

In this paper, we will prove global well-posedness for the Cauchy problems of two modified-Leray-a-MHD models withpartial viscous terms. Copyright © 2009 John Wiley & Sons, Ltd.

Keywords: modified-Leray-�-MHD model; uniqueness; partial viscosity

1. Introduction

First, we consider the following modified-Leray-�-MHD model [1]:

vt +(v ·∇)u−��v+∇�+ 12 ∇|B|2 = (B ·∇)B (1)

Bt +(u ·∇)B−(B ·∇)u−��B = 0 (2)

v = (1−�2�)u (3)

div v = div u=div B=0 (4)

(v, B)|t=0 = (v0, B0) in R2 (5)

where v is the fluid velocity field, u is the ‘filtered’ fluid velocity, B is the magnetic field and � is the pressure, are the unknowns;� is the viscosity and � is the magnetic diffusivity, � is the lengthscale parameter that represents the width of the filter, and forsimplicity, we will take �≡1.

The existence and uniqueness of weak solutions have been proved in [1]. When B≡0, the above system has been studied in [2].When �→0, the above system reduces to the well-known MHD equations, which have received many studies in the recent years(Fan and Zhou, submitted) [3--10].

In this paper, we will establish a global well-posedness for (1)–(5) with �=0.

Theorem 1.1(n=2,�=1,�=0). Let (v0, B0)∈H3 ×H3 and v0 = (1−�)u0, with div v0 =div u0 =div B0 =0 in R2. Then the problem (1)–(5) has aunique global smooth solution (v, B) such that

v ∈L∞(0, T; H3)∩L2(0, T; H4), B∈L∞(0, T; H3) for any T>0

aDepartment of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People’s Republic of ChinabDepartment of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, People’s Republic of China∗Correspondence to: Yong Zhou, Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People’s Republic of China.†E-mail: yzhoumath@zjnu.cn

Contract/grant sponsor: New Century Excellent Talents in Universities in China; contract/grant number: NCET-07-0299Contract/grant sponsor: Shuguang Project; contract/grant number: 07SG29Contract/grant sponsor: Shanghai Rising Star Program; contract/grant number: 08QH14006Contract/grant sponsor: Fok Ying Tong Education Foundation; contract/grant number: 111002

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Remark 1.1At present, we are unable to establish global well-posedness as n=2, �=0 and �=1. It is reasonable (in some sense), since it iswell-known that the velocity plays a dominant role in the MHD-type equations.

Next, we consider another modified-Leray-�-MHD model:

vt +(u ·∇)u−��v+∇�+ 12 ∇|B|2 = (B ·∇)B (6)

Bt +(u ·∇)B−(B ·∇)u−��B = 0 (7)

v = (1−�2�)u (8)

div v = div u=div B=0 (9)

(v, B)|t=0 = (v0, B0) in Rn (n=2, 3) (10)

When B≡0, the system (6) has been studied in [11] with �>0 or �=0. When �→0, the above system (6)–(9) also reduces to thewell-known MHD equations. By some similar argument as that in [1], it is easy to prove the global well-posedness with �>0 and�>0, hence we omit the detail for concision.

Comparing with these two models, one can find easily that the second one is nothing but the first one with one term u ·∇uinstead of (u−�u) ·∇u. Hence, roughly speaking, the second one is simpler than the first one. Hence, a similar theorem as that forthe first model holds.

Theorem 1.2(n=2,�=1,�=0). Let (v0, B0)∈H3 ×H3 and v0 = (1−�)u0, with div v0 =div u0 =div B0 =0 in R2. Then the problem (6)–(10) has aunique global smooth solution (v, B) such that

v ∈L∞(0, T; H3)∩L2(0, T; H4), B∈L∞(0, T; H3) for any T>0

Since the proof of Theorem 1.2 is similar to and simpler than that of Theorem 1.1, we omit the details in this paper for concision.Moreover, a more interesting thing is for the case with partial viscosity (�=0,�=1). We will prove.

Theorem 1.3(n=3,�=0,�=1). Let (v0, B0)∈H3 ×H3 and v0 = (1−�)u0, with div v0 =div u0 =div B0 =0 in R3. Then the problem (6)–(10) has aunique global smooth solution (v, B) such that

v ∈L∞(0, T; H3), B∈L∞(0, T; H3)∩L2(0, T; H4) for any T>0

2. Proof of Theorem 1.1

It is sufficient to establish a priori estimates for any T .First, multiplying (1) and (2) by u and B, respectively, after integration by parts, summing up them, we easily get

1

2

∫u2 +|∇u|2 +B2 dx+

∫ T

0

∫|∇u|2 +|�u|2 dx dt�1

2

∫u2

0 +|∇u0|2 +B20 dx (11)

where divergence free of u and B gives ∫(u ·∇u) ·u dx = −

∫(u ·∇u) ·u dx =0

∫ (∇�+ 1

2∇B

)·u dx =

∫ (�+ 1

2B

)·(div u) dx =0

and ∫(u ·∇B) ·B dx =0,

∫(B ·∇B) ·u+(B ·∇u) ·B dx =0

Hence, ∫ T

0

∫|v|2 dx dt�C (12)

Applying curl to (1) and setting � :=curl v give

�t −��=B ·∇curl B−v∇curl u+ �v

�y∇u1 − �v

�x∇u2 (13)

where u1 and u2 are two components of u.

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Multiplying (13) by �, using div B=0, div v =0, curl∇ ≡0, and using the theory of Hardy space in Coifman–Lions–Meyer–Semmes[12], we infer that

1

2

d

dt

∫�2 dx+

∫|∇�|2 dx =

∫(B ·∇curl B)�dx−

∫(v ·∇curl u) ·�dx+

∫ (�v

�y∇u1 − �v

�x∇u2

)�dx

� ‖B ·∇curl B‖H1 ·‖�‖BMO +‖v ·∇curl u‖H1 ·‖�‖BMO

+∥∥∥∥�v

�y∇u1

∥∥∥∥H1

·‖�‖BMO +∥∥∥∥�v

�x∇u2

∥∥∥∥H1

·‖�‖BMO

� C‖B‖L2 ·‖∇curl B‖L2 ·‖∇�‖L2 +C(‖v‖L2‖∇curl u‖L2 +‖∇v‖L2 ·‖∇u‖L2 )‖∇�‖L2

� C‖�B‖L2‖∇�‖L2 +C(‖v‖2L2 +‖�‖L2 )‖∇�‖L2

� C‖�B‖L2‖∇�‖L2 +C‖�‖L2‖∇�‖L2

� 1

8‖∇�‖2

L2 +C‖�B‖2L2 +C‖�‖2

L2 (14)

In the following estimates, we will use the following commutator estimates and bilinear estimates due to Kato and Ponce [13]and Kenig et al. [14], respectively:

‖��(fg)−f��g‖Lp � C(‖∇f‖Lp1 ‖��−1g‖Lq1 +‖��f‖Lp2 ‖g‖Lq2 ) (15)

‖��(fg)‖Lp � C(‖f‖Lp1 ‖��g‖Lq1 +‖��f‖Lp2 ‖g‖Lq2 ) (16)

with �>0, � := (−�)1/2 and

1

p= 1

p1+ 1

q1= 1

p2+ 1

q2

Taking � on (2), multiplying it by �B, using (15) and (16), we find that

1

2

d

dt

∫|�B|2 dx �

∣∣∣∣∫

(�(u ·∇B)−u∇ ·�B) ·�B dx

∣∣∣∣+∣∣∣∣∫

�(B ·∇u) ·�B dx

∣∣∣∣� C‖∇u‖L∞ ·‖�B‖2

L2 +C‖�u‖L4 ·‖∇B‖L4 ·‖�B‖L2

+C‖B‖L∞ ·‖∇�u‖L2 ·‖�B‖L2 =: I1 + I2 + I3 (17)

We will use the following logarithmic Sobolev inequality due to Brezis and Gallouet [15] and Brezis and Wainger [16] (see alsoEngler [17] and Ozawa [18]):

‖f‖L∞�C‖f‖H1 log(e+‖f‖H2 ) (18)

to estimate I1 as follows:

I1 � C‖u‖H2 log(e+‖u‖H3 ) ·‖�B‖2L2

� C‖v‖L2 log(e+‖�‖L2 +‖�B‖L2 )‖�B‖2L2

I2 � C‖v‖L4 ·‖∇B‖L4 ·‖�B‖L2

� C‖v‖3/4L2 ‖�v‖1/4

L2 ·‖B‖1/4L2 ‖�B‖3/4

L2 ·‖�B‖L2

� C‖v‖3/4L2 ‖∇�‖1/4

L2 ·‖�B‖7/4L2

� 14 ‖∇�‖2

L2 +C‖v‖6/7L2 ‖�B‖2

L2

where we used the following Gagliardo–Nirenberg inequalities:

‖v‖L4 � C‖v‖3/4L2 ‖�v‖1/4

L2 (19)

‖∇B‖L4 � C‖B‖1/4L2 ‖�B‖3/4

L2 (20)

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While I3 can be estimated as

I3 � C‖B‖L∞ ·‖∇v‖L2 ·‖�B‖L2

� C‖B‖1/2L2 ‖�B‖1/2

L2 ·‖v‖1/2L2 ‖�v‖1/2

L2 ·‖�B‖L2

� C‖�B‖3/2L2 ·‖v‖1/2

L2 ·‖∇�‖1/2L2

� 14 ‖∇�‖2

L2 +C‖v‖2/3L2 ‖�B‖2

L2

here the following Gagliardo–Nirenberg inequalities were used:

‖B‖L∞ � C‖B‖1/2L2 ‖�B‖1/2

L2 (21)

‖∇v‖L2 � C‖v‖1/2L2 ‖�v‖1/2

L2 (22)

Inserting the above estimates into (17) and combining (14), using (11) and (12), we conclude that

‖�‖L∞(0,T;L2) +‖�‖L2(0,T;H1) � C (23)

‖B‖L∞(0,T;H2) � C (24)

Therefore, we have

‖v‖L∞(0,T;H1) +‖v‖L2(0,T;H2) � C (25)

‖u‖L∞(0,T;H3) +‖u‖L2(0,T;H4) � C (26)

Applying �3 to (2), multiplying it by �3B, using (15), (16), (26) and the Sobolev inequality, we have

1

2

d

dt

∫|�3B|2 dx �

∣∣∣∣∫

(�3(u ·∇B)−u ·∇�3B) ·�3B dx

∣∣∣∣+∣∣∣∣∫

�3(B ·∇u) ·�3B dx

∣∣∣∣� C‖∇u‖L∞‖�3B‖2

L2 +C‖∇B‖L∞ ·‖�3u‖L2 ·‖�3B‖L2 +C‖B‖L∞ ·‖�4u‖L2 ·‖�3B‖L2

� C‖�3B‖2L2 +C‖∇B‖L∞ ·‖�3B‖L2 +C‖�4u‖L2‖�3B‖L2

� C‖�3B‖2L2 +C(1+‖�3B‖L2 )‖�3B‖L2 +C‖�4u‖L2‖�3B‖L2

which yields

‖B‖L∞(0,T;H3)�C (27)

by Gronwall’s inequality.Taking � to (13), multiplying it by ��, using (16), and (23)–(27), we have

1

2

d

dt

∫|��|2 dx+

∫|∇��|2 dx �

∣∣∣∣∫

∇(v ·∇curl u) ·∇��dx

∣∣∣∣+∣∣∣∣∫

∇(

�v

�y∇u1 − �v

�x∇u2

)∇��dx

∣∣∣∣

+∣∣∣∣∣

2∑i=1

∫��i(BiB) ·curl��dx

∣∣∣∣∣� C‖v‖L4‖�‖L4‖∇��‖L2 +C‖∇u‖L∞ ·‖∇�‖L2 ·‖∇��‖L2

+C‖B‖L∞ ·‖�3B‖L2 ·‖∇��‖L2

� C‖�‖L4‖∇��‖L2 +C‖∇�‖L2‖∇��‖L2 +C‖∇��‖L2

� 1

2‖∇��‖2

L2 +C‖∇�‖2L2 +C‖��‖2

L2 +C (28)

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Y. ZHOU AND J. FAN

which implies

‖�‖L∞(0,T;H2) +‖�‖L2(0,T;H3)�C (29)

Hence

‖v‖L∞(0,T;H3) +‖v‖L2(0,T;H4)�C (30)

Owing to Equation (3), we have the following regularity for u:

u∈L∞(0, T; H5)∩L2(0, T; H6)

Taking div-operator on (1), it is easy to find that the pressure � is solved by the following equation:

−��= 12 �|B|2 +div (u ·∇u)−div (B ·∇B)

This completes the proof.

3. Proof of Theorem 1.2

First, we also have (11) and (12).Multiplying (7) by |B|2B, using (9), (11) and setting H :=B2, we have

1

4

d

dt

∫H2 dx+ 3

4

∫|∇H|2 dx =

∫(B ·∇)u ·B2B dx�

∫|∇u|·H2dx�‖∇u‖L2‖H‖2

L4

� C‖H‖2L4�C‖H‖1/2

L2 ·‖∇H‖3/2L2

� 1

2‖∇H‖2

L2 +C‖H‖2L2

which yields

‖B‖L∞(0,T;L4) +‖B ·∇B‖L2(0,T;L2)�C (31)

Here we have used the following Gagliardo–Nirenberg inequality:

‖H‖L2p/ (p−2)�C‖H‖1−3/pL2 ‖∇H‖3/p

L2 (p>3) (32)

Multiplying (6) by v, using (9), (11), (31) and (32), after integration by parts, we find that

1

2

d

dt

∫v2 dx =

∫(B ·∇)B ·v dx−

∫(u ·∇)u ·v dx

� ‖B ·∇B‖L2‖v‖L2 +‖u‖L6‖∇u‖L3‖v‖L2

� ‖B ·∇B‖L2‖v‖L2 +C‖∇u‖3/2L2 ·‖�u‖1/2

L2 ‖v‖L2

� ‖B ·∇B‖L2‖v‖L2 +C‖v‖3/2L2

which gives

‖v‖L∞(0,T;L2) +‖u‖L∞(0,T;H2)�C (33)

Taking � to (7), multiplying it by �B, using (33), we obtain that

1

2

d

dt

∫|�B|2 dx+

∫|∇�B|2 dx =

∫�(B ·∇u−u ·∇B) ·�B dx =−

∫∇(B ·∇u−u ·∇B) ·∇�B dx

� C

∫(|B|·|�u|+|∇B|·|∇u|+|u|·|�B|)|∇�B|dx

� C(‖B‖L∞‖�u‖L2 +‖∇u‖L6‖∇B‖L3 +‖u‖L∞‖�B‖L2 )‖∇�B‖L2

� C(‖B‖L∞ +‖∇B‖L3 +‖�B‖L2 )‖∇�B‖L2

� C‖B‖H2‖∇�B‖L2�1

2‖∇�B‖2

L2 +C‖B‖2H2

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Copyright © 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2010, 33 856–862

Y. ZHOU AND J. FAN

which implies

‖B‖L∞(0,T;H2) +‖B‖L2(0,T;H3)�C (34)

Applying � to (6), multiplying it by �v, using (9), (8), (16), (34) and (33), we infer that

1

2

d

dt

∫|�v|2 dx �

∫�(B ·∇B−u ·∇u) ·�v dx

� C(‖B‖L∞‖B‖H3 +‖u‖L∞‖u‖H3 )‖�v‖L2

� C(‖B‖H3 +‖u‖H3 )‖�v‖L2

� C(‖B‖H3 +‖v‖H1 )‖�v‖L2

which leads to

‖v‖L∞(0,T;H2) +‖u‖L∞(0,T;H4)�C (35)

Taking �3 to (7), multiplying it by �3B, using (16), (35) and (34), we conclude that

1

2

d

dt

∫|�3B|2 dx+

∫|∇�3B|2 dx = −

∫�3(u ·∇B−B ·∇u) ·�3B dx =

∫�2(u ·∇B−B ·∇u) ·�4B dx

� C(‖u‖L∞‖B‖H3 +‖B‖L∞‖u‖H3 )‖�4B‖L2

� C(‖B‖H3 +1)‖�4B‖L2

which tells us that

‖B‖L∞(0,T;H3) +‖B‖L2(0,T;H4)�C (36)

Taking �3 to (6), multiplying it by �3v, using (9), (8), (16), (35) and (36), we arrive at

1

2

d

dt

∫|�3v|2 dx �

∫�3[(B ·∇)B−(u ·∇)u] ·�3v dx

� C(‖B‖L∞‖B‖H4 +‖u‖L∞‖u‖H4 )‖�3v‖L2

� C(‖B‖H4 +‖u‖H4 )‖�3v‖L2

� C(1+‖B‖H4 )‖�3v‖L2

which shows that

‖v‖L∞(0,T;H3)�C (37)

This finishes the proof.

Acknowledgements

The authors thank the referee for his/her helpful suggestions. This work is partially supported by Program for New Century ExcellentTalents in Universities in China (No. NCET-07-0299), Shuguang Project (No. 07SG29), Shanghai Rising Star Program (No. 08QH14006)and Fok Ying Tong Education Foundation (No. 111002).

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